Robust wind farm layout optimization using pseudo-gradients
Erik Quaeghebeur
1 Wind energy systemsA wind energy system transforms wind into electrical power.
1.1 Wind turbines
© Hans Hillewaert / CC BY-SA 4.0
Wind turbines (picture left) are the elementary wind energy sys-tems. Important characteristics are its rated power, rotor diame-ter, and hub height.
v0 vcut-in vcut-out
p
rated power
ct‘maximum’ thrustA high-level modelconsists of the powercurve and the thrustcurve, which map windspeed at hub heightto power and forceexerted on the wind(plot above right).
1.2 Wind farms
© Thomas Nugent / CC BY-SA 2.0
Wind farms are collections ofwind turbines constrained toa specific site (picture left).
The placement of turbines within a farmis its layout (drawing right).
The layout influences the farm cost via thecabling and substructure cost, due to cablelayout and depth & soil variations.
1.3 Wake lossesWakes are regions of complexly perturbed wind behind turbinerotors (picture right).
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Computationally simple engi-neering wake models are usedwhen calculating farm poweroutput (simulation below right).
In a farm, wakes may reducethe wind speed at downstreamturbines, causing lower powerproduction: wake losses. Wakewind speed deficits for a givenlayout depend on the wind direc-
tion (plot above, corresponds to layout shown in Section 1.2).
© Vattenfall
© Harms et al. (2016 student project)
2 Wind resourcesA wind resource is the wind available at a wind farm site.
2.1 Wind direction & speed distributions0°
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The minimal wind re-source description re-quired is a joint winddirection & speed dis-tribution (plot left);there is a dependencybetween bothcomponents.
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This joint is decomposed into the wind rose, thewind direction marginal (plot above right), and per-direction wind speed conditionals (plot right), forwhich Weibull distributions are often used.
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2.2 Annual energy production of a wind farm
An essential quantity in the design of a wind farm is its annualenergy production (AEP): the electrical energy produced bya farm for a given wind resource.
Equivalent is the capacity factor, the ratio between the ex-pected average power production and the farm’s rated power.
Also of interest is the power rose, the distribution over winddirections of relative wakeless power production (plot right).
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2.3 Inter-year wind resource variation0°
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0.140.16 We consider 35 yearly wind re-
sources for a North Sea site fromthe Dutch meteorological institute’s‘KNW atlas’ (plot left: orange lower,blue average, and red upper windroses for this set of distributions; plotright: corresponding power roses).
Note the substantial variation.
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4 Inter-year variation robustnessA wind farm’s layout is usually optimized for one wind resource, the estimated average oneover the farm lifetime. However, inter-year production stability is important for the finan-cial attractiveness of a farm design. Making a farm robust against inter-year wind re-source variation is therefore of practical interest.
4.1 Goals• Quantify inter-year wind resource
variation (done).• Quantify inter-year AEP variation
(done).• Determine existence of robust
farm layouts (partly done).• Develop robust layout optimiza-
tion algorithm (not yet done).
4.2 Setup• Realistic test site.• Realistic & extensive set of yearly wind resources.• Create optimized layout for
– a degenerate wind resource (‘225°’),– the uniform wind resource,– each wind resource in the set,– their average,– their lower & upper envelopes.
4.3 Results
2.8 3.0 3.2 3.4 3.6 3.8 4.0Wake loss percentage
19791980198119821983198419851986198719881989199019911992199319941995199619971998199920002001200220032004200520062007200820092010201120122013
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year-optimizedupperall yearsloweruniformonly 225°
4.4 Conclusions• Inter-year variation is substantial.• Observed inter-year variation is larger
than inter-layout differences.• The set of layouts with undominated
production profiles is relatively small.• No real trade-off achieved yet between
robustness and optimality.
4.5 Recommendations• Create a more diverse set of layouts:
– by varying the optimizer parameters,– by using different optimization algorithms.
• Try out ideas for robust optimization:– by each iteration using the maximin solution
over wind resources,– by following your suggestion.
3 Wind farm layout optimization
3.1 Objectives• AEP: Maximize for expected power production only
(used in our study).• LCoE: Minimize levelized cost of energy,
the ratio between farm cost and power production (more realistic).
3.2 ConstraintsTurbines in a farm must satisfy a distance constraint (drawing right,red circles) and site constraints (drawing right, red & blue lines).
3.3 Typical layout optimization algorithmstype gradient-based heuristic (usually random search-based)
examples steepest ascent evolutionary, genetic, particle swarmpros high-quality solutions flexible (generic)cons computationally expensive, computationally expensive,
can get stuck in local optima, does not use domain knowledge,problem-specific preparation low-quality solutions
Computational cost is crucial in robustness studies, so we developed a fast heuristicapproach that uses domain knowledge and produces medium-quality solutions.
3.4 Pseudo-gradient-based optimization0°
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1.0For one wind direction (plot left),the power deficit of a down-stream turbine due to an upstreamone determines a vector. Averageover all upstream turbines (plot right).
Variant: vectors pushing upstreamturbines ‘back’ (plot far right).
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Taking the expectation over all direc-tions (plot left) gives ‘pseu-do-gradients’ usable in a localgradient ascent-type algorithm.
Applicable to all variants (plot right:‘push down’; plot far right: ‘push back’).
We have created a layout optimization algorithm that each iteration:
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0.0550• uses an adaptive step size,• considers pseudo-gradients for
each of the variants,• greedily moves turbines
according to the best one, and• corrects constraint violations be-
tween steps by iteratively movingturbines to satisfying positions.
We obtain good convergence (plot above left, relative wake loss) andmedium-quality layouts (drawing above right, turbine trajectories).
